ICTS

This lecture is composed of three chapters. In the first, I will first present the basic ingredients of the mechanics of plates and shells, and then the geometrical properties of these two dimensional objects, which play a crucial role in shaping their response to external loads. In the second chapter, I introduce pattern formation in the form of wrinkles, folds and blisters. I start to describe them in a one-dimensional setting, and then discuss how additional effects may come into play in two-dimensional configurations. Last, I will present reduced models aiming at describing the global shape of the plates when they buckle and the extent and properties of the wrinkle patterns. The third chapter will discuss the phenomenon of stress focusing through elementary stress focusing patterns: the conical and ridge singularities. Last, I will discuss "crumples", a pattern that is quite common yet not fully understood.

Unlike motile animal cells driven by cytoskeletal forces, plant cells are confined by rigid walls and cannot migrate. Their development thus relies on cell growth and division, governed by hydraulic processes.

Plant growth is a hydromechanical phenomenon: cells absorb water osmotically, remodel their walls, and expand irreversibly, generating tissue-level deformations and form. In this first part of this series of lecture on the mechanics and mathematics of plant morphogenesis, we focus on individual plant cells, exploring the fundamental mechanisms of cell expansion and their mathematical modeling. Building on Lockhart’s (1965) classical model, we introduce the concept of turgor pressure and its link to growth, mechanics, and geometry—foundations for a theory of plant active matter.

To survive and thrive, plants depend on their ability to sense and integrate multiple environmental cues—such as gravity, light, and touch—and to translate these signals into growth and shape change. For this to occur, external stimuli must be transmitted down to the cellular scale, where they generate the physical deformations that drive morphogenesis. To capture this process from first principles, we propose a multiscale theory of tropism that links environmental stimuli to hormone transport, which in turn governs tissue-level growth and remodeling. This coupling between signal perception, transport, and deformation dynamically adjusts the plant’s form and orientation in response to its environment. The framework can be extended to explore not only individual tropic responses but also the emergent behaviors arising from multiple, interacting, and evolving stimuli.
 

In these lectures we will learn the intrinsic approach to the mechanics of growing and self shaping assemblies. In contrast to the configuration-based standard mechanical approach, in the the intrinsic approach we consider the local shaping tendencies of a body. We will discovery why this approach is particularly useful for studying growing tissue, and discuss the ubiquity and role of geometric frustration in these structures.